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thermal conductivity and groundwater flux in porous
media
Nataline Simon, Olivier Bour, Nicolas Lavenant, Gilles Porel, Benoît Nauleau,
Behzad Pouladi, Laurent Longuevergne, Alain Crave
To cite this version:
Nataline Simon, Olivier Bour, Nicolas Lavenant, Gilles Porel, Benoît Nauleau, et al.. Numerical and
experimental validation of the applicability of active-DTS experiments to estimate thermal
conductiv-ity and groundwater flux in porous media. Water Resources Research, American Geophysical Union,
2021, 57 (1), pp.e2020WR028078. �10.1029/2020WR028078�. �insu-03048099�
1. Introduction
In groundwater hydrology, the characterization of the distribution of groundwater flow within the critical zone received considerable attention in the last decades (Freeze & Cherry, 1979). Our ability to quantify groundwater flow greatly controls our ability to characterize aquifers, predict contaminant transport, and understand biogeochemical reactions and processes occurring in the subsurface (Kalbus et al., 2009; Poeter & Gaylord, 1990). Groundwater flow at interfaces such as recharge and discharge areas also plays a key role in the preservation of groundwater-dependent ecosystems (Kalbus et al., 2006; Sophocleous, 2002). The quantification of groundwater fluxes is also particularly relevant for geothermal energy since they control heat exchange and storage capacities (Diao et al., 2004). Similarly, the characterization of seepage through dams, dikes, and reservoirs is also critical for geotechnical engineering (Foster et al., 2000).
The spatial distribution of groundwater fluxes is largely driven by subsurface heterogeneities. Thus, in past decades, the characterization of the distribution of groundwater fluxes and their quantification relied on the capacity of characterizing and modeling the spatial variability of hydraulic conductivities (de Marsi-ly, 1976). Considering the challenge in characterizing the field variability of hydraulic properties, the use of heat as a tracer has been widely developed and applied to characterize flow in aquifers or at interfaces such as the hyporheic zone (Anderson, 2005; Kurylyk & Irvine, 2019; Kurylyk et al., 2019; Rau et al., 2014). In-deed, the heat propagation is particularly sensitive to flow variations because thermal properties are much less variable than hydraulic conductivity (Anderson, 2005; Mao et al., 2013). Thus, the thermal conductivity
Abstract
Groundwater flow depends on the heterogeneity of hydraulic properties whose field characterization is challenging. Recently developed active-Distributed Temperature Sensing (DTS) experiments offer the possibility to directly measure groundwater fluxes resulting from heterogeneous flow fields. Here, based on fundamental principles and numerical simulations, two interpretation methods of active-DTS experiments are proposed to estimate both the porous media thermal conductivities and the groundwater fluxes in sediments. These methods rely on the interpretation of the temperature increase measured along a single heated fiber-optic (FO) cable and consider heat transfer processes occurring both through the FO cable itself and through the porous media. The first method relies on the Moving Instantaneous Line Source model that reproduces the temperature increase and provides estimates of thermal conductivity and groundwater flux as well as an evaluation of the temperature rise due to the FO cable. The second method, based on the graphical identification of three characteristic times, provides complementary estimates of flux, fully independent of the effect of the FO cable. Sandbox experiments provide an experimental validation of the interpretation methods, demonstrate the excellent accuracy of groundwater flux estimates (<5%), and highlight the complementarity of both methods. Active-DTS experiments allow investigating groundwater fluxes over a large range spanning 1 × 10−6−5 × 10−2 m/s,depending on the duration of the experiment. Considering the applicability of active-DTS experiments in different contexts, we propose a general experimental framework for the application of both interpretation methods in the field, making active-DTS field experiments especially promising for many subsurface applications.
© 2020. American Geophysical Union. All Rights Reserved.
Thermal Conductivity and Groundwater Flux in Porous
Media
N. Simon1 , O. Bour1, N. Lavenant1, G. Porel2 , B. Nauleau2, B. Pouladi1, L. Longuevergne1 , and A. Crave1
1Univ Rennes, CNRS, Géosciences Rennes – UMR 6118, Rennes, France, 2Department of Earth Sciences, IC2MP UMR
7285, Université de Poitiers, CNRS, HydrASA, Poitiers, France
Key Points:
• Numerical and experimental validation of two methods to interpret active-Distributed Temperature Sensing (DTS) experiments in sediments • Determination of thermal
conductivity and groundwater flux with low uncertainties
• Definition of the applicability, range of measurements, and limits of active-DTS experiments
Correspondence to:
N. Simon and O. Bour
nataline.simon@univ-rennes1.fr;
olivier.bour@univ-rennes1.fr
Citation:
Simon, N., Bour, O., Lavenant, N., Porel, G., Nauleau, B., Pouladi, B., et al. (2021). Numerical and experimental validation of the applicability of active-DTS experiments to estimate thermal conductivity and groundwater flux in porous media. Water Resources
Research, 57, e2020WR028078. https:// doi.org/10.1029/2020WR028078
Received 5 JUN 2020 Accepted 28 NOV 2020
ranges between 0.9 and 4 W/m/K for sedimentary aquifers (Stauffer et al., 2013) while hydraulic conductiv-ity can vary over 12 order of magnitude (Freeze & Cherry, 1979).
Fiber-Optic-Distributed Temperature Sensing (FO-DTS) technology provides continuous temperature re-cords through space and time along fiber-optic (FO) cables at high spatial and temporal resolutions (Habel et al., 2009; SEAFOM, 2010; Smolen & van der Spek, 2003; Tyler et al., 2009; Ukil et al., 2012). The best performing DTS units available (e.g., Silixa Ultima) can provide temperature measurements every second at a 0.125-m sample spacing. Accuracy of temperature measurements and effective spatial resolution de-pends not only on the performance of the DTS units but also on integration time, FO cable selection, and experimental conditions in the field (Simon et al., 2020). Many applications in hydrologic sciences demon-strated the potential of the tool to characterize water movements and distribution in the subsurface (Selker et al., 2006a, 2006b ; Shanafield et al., 2018), especially since the development of active-DTS methods (Bense et al., 2016). Active-DTS methods continuously monitor temperature changes, induced by a heat source and measured along a FO cable. Early applications in open boreholes demonstrated the capability of heat tracer tests to quantify borehole flows (Banks et al., 2014; Leaf et al., 2012; Liu et al., 2013; Read et al., 2015). Contrary to passive-DTS methods that require measuring natural thermal anomalies or natural temperature fluctuations over time to characterize groundwater flows (Anderson, 2005), active-DTS methods are more sensitive to flow and allow investigating flow variations independently of temperature anomalies.
Among active-DTS methods, some authors proposed to monitor the difference of temperature measured between an electrically heated and a nonheated FO cable, which is directly dependent on fluxes (Bense et al., 2016; Read et al., 2014; Sayde et al., 2015). This approach offered the possibility to determine flow velocity distribution all along the FO cable and to quantify fluxes at unprecedented spatial and temporal resolutions. Then, considering the need to characterize flow under a natural gradient without the effect of the borehole (Pehme et al., 2010), further developments focused on deploying active-DTS methods in direct contact with the rock matrix. Active-DTS experiments have thus been conducted in sealed bore-holes (Coleman et al., 2015; Maldaner et al., 2019; Munn et al., 2020; F. Selker & J. S. Selker, 2018) and other promising studies proposed the direct deployment of FO cables vertically into unconsolidated sedi-mentary aquifers using direct-push equipment (Bakker et al., 2015; des Tombe et al., 2019). Concurrently, active-DTS methods were largely developed and applied in unsaturated soils, offering the possibility of estimating the soil water content and thermal properties (Benitez-Buelga et al., 2014; He, Dyck, Horton, Li, et al., 2018; He, Dyck, Horton, Ren, et al., 2018; Sayde et al, 2010, 2014; Weiss, 2003; Wu et al., 2019) or con-ducting distributed thermal response test for geothermal energy applications (Vélez Márquez et al., 2018; Zhang et al., 2020).
When heating is applied in direct contact with a saturated porous media, the thermal response, namely the elevation of temperature measured along the cable during heating periods, depends on thermal pro-cesses through the porous media and thus on water fluxes that dissipate heat through advection. Differ-ent approaches have been proposed to interpret this thermal response, such as numerical models (Cole-man et al., 2015; Maldaner et al., 2019) and empirical or analytical solutions (Aufleger et al., 2007; Bakx et al., 2019; Maldaner et al., 2019; Perzlmaier et al., 2004). However, the required parameter calibration often makes the application of models difficult in different contexts. Bakker et al. (2015) and des Tombe et al. (2019) have recently proposed the use of analytical solutions, initially developed by Carslaw and Jae-ger (1959), to efficiently estimate the distribution of fluxes along the FO cable. These solutions explicitly take into account conduction and advection processes occurring in saturated porous media. However, these applications did not consider the possible spatial variations of the thermal conductivity of sediments. More-over, since their setup uses a heating cable combined with a separate FO cable used to monitor temperature at a given distance, their analytical solution cannot therefore be applied directly when the same cable is used both as a heat source and to monitor the temperature (Del Val, 2020).
Although previous studies showed the link between groundwater flow and heat dissipation, most appli-cations still depend on a calibration process or involve data uncertainties that preclude the generalization or full validation of active-DTS experiments applied to saturated sediments. To summarize, the use of ac-tive-DTS methods to quantify groundwater fluxes in the subsurface would require (i) a general interpreta-tion method that can be easily applied and that takes into account the spatial distribuinterpreta-tion of the thermal conductivities of sediments, (ii) a validation of the methodology with independent experimental data, and
(iii) a clear discussion about the applicability of the method in the field considering both the range of fluxes that can be measured and the limits of the method.
These are the aims of the present study. In the following, we first present the theoretical background re-quired to interpret active-DTS experiments before presenting the numerical model used to simulate ac-tive-DTS experiments in porous media as well as the experiments achieved in the sandbox. Then, numerical simulations results provide an improved understanding of thermal processes controlling the temperature increase. Two interpretation methods are then proposed to interpret the temperature increase measured along a single heated FO cable during active-DTS experiments, providing an estimate of the spatial distri-bution of both groundwater fluxes and thermal conductivities at an unprecedented uncertainty level. These interpretation methods are first tested by numerical simulations before being experimentally validated. Finally, we discuss the applicability of active-DTS experiments by defining the limitations, range, and un-certainties of measurements. This study offers the possibility of generalizing the application of active-DTS experiments in the field, for a wide range of contexts.
2. Materials and Methods
2.1. Theoretical Background
2.1.1. Heat Transfer Through Porous Media
To quantify groundwater fluxes, we propose deploying a heat source within a porous media and monitoring the surrounding temperature evolution through an active heat tracer experiment using FO-DTS technology. The cable is heated electrically through its steel armoring while the elevation in temperature is continu-ously monitored using the fiber optic inside the cable. Therefore, a single FO cable is used to measure the surrounding temperature and as a heat source, as applied in some previous studies (Bakx et al., 2019; Bense et al., 2016; Read et al., 2014; F. Selker & J. S. Selker, 2018; Simon et al., 2020).
In this configuration, the temperature increase measured during an active-DTS experiment should be con-trolled by heat transfer processes occurring in the porous medium. Within a saturated porous medium, heat is transferred by two main mechanisms: advection, which corresponds to the heat transferred by flowing groundwater; and conduction, which corresponds to the molecular diffusion of thermal energy (Ander-son, 2005). Without any flow, heat transfer should occur only by conduction and a gradual and continuous increase of temperature is therefore expected. If water flows through the porous medium, advection should partly control the thermal response by dissipating the heat produced by the heat source. The higher the water flow velocity, the lower should be the temperature increase. Note that free thermal convection in the porous media is considered here negligible, which is one of the assumptions required applying the follow-ing analytical solutions.
The transport of heat in saturated porous media by conduction and advection is described by the following heat transport equation (Carslaw & Jaeger, 1959; Domenico & Schwartz, 1998):
2 2 2 2 w w t T D T T q c T t x y c x (1) Equation 1 corresponds to the 2D advection-diffusion equation for a homogeneous and isotropic porous medium with a uniform and constant fluid flux q in x direction as proposed by Stallman (1965). In this equation, T is the temperature (K), q the groundwater flux (or specific discharge) (m/s), ρc the volumetric
heat capacity of the rock–fluid matrix (J/m3/K), and ρ
wcw the volumetric heat capacity of water (J/m3/K). Dt is the thermal diffusivity coefficient (m2/s) and depends on λ, the bulk thermal conductivity (W/m/K):
t D c (2)
2.1.2. The Instantaneous Line Source Model
As highlighted by the heat transport equation, the temperature evolution is controlled by both the ground-water flux and the thermal properties of the saturated material, especially the thermal conductivity. Under
no-flow conditions (q = 0 in Equation 1), Carslaw and Jaeger (1959) showed that the differential equation for heat conduction could be solved for a homogeneous infinite medium, initially at thermal equilibrium (T0 everywhere) by considering an Instantaneous Line Source (ILS). At t = 0, a constant linear heating
pow-er Q (W/m) is switched on. The temppow-erature response ΔT (ΔT = T – T0) is given in x–y direction by
”T x y t Q d Q E r tD r tDt i t , , exp
4 2 4 4 4 2 (3) where ²r x² y² when the heat source is located at (0, 0), ΔT the change of temperature across the input time, and Ψ a change of variable:
x y
D tt
² ²
4
(4) The exponential integral function –Ei
1x
is also known in hydrogeology as the Well-function W(x). Thus, Equation 3 can be simplified by the Jacob approximation if t is sufficiently large or r very small, i.e., if r2/4Dtt << 1 (Blackwell, 1954; de Marsily, 1976): 2 2 4 2.25 Δ ln 4 t 4 t Q tD Q t D T W r r (5) This assumption was used by Blackwell (1954) who developed a transient-flow method showing that the thermal conductivity λ could be extracted from the slope s of the linear regression between ΔT and ln(t) during the heating period:
22
11 d dln ln ln 4 T t T t T Q s t t t (6) Recent applications of active-DTS methods in boreholes used this approach to estimate the distribution of thermal conductivities (Freifeld et al., 2008; Maldaner et al., 2019). However, the estimation of thermal conductivity in active-DTS applications achieved under flow conditions remains challenging due to the combined effects of both thermal conductivity and groundwater flux on the temperature increase.2.1.3. The Moving Instantaneous Line Source Model
Carslaw and Jaeger (1959) proposed adapting the ILS model considering uniform flow across the heat source. They proposed solving the advection–conduction equation by considering a Moving Instantaneous Line Source (MILS) model that includes the effect of advection on the thermal response. By considering an initial thermal equilibrium T0 and a constant and uniform heating rate power Q (W/m), the thermal
response ΔT (ΔT = T – T0) along the line source can be expressed as
T x y Q q x D c c x y D t w w x y tDt t , exp exp ²
4 2 2 4 2 2 q D c c d t w w 2 2 16 ² ² ² (7a) Different applications showed the potential of using this analytical solution to solve the advection–conduc-tion equaadvection–conduc-tion for uniform and isotropic materials (Sutton et al., 2003; Zubair & Chaudhry, 1996). It is note-worthy that this solution can also be used for anisotropic material considering thermal dispersion (Carslaw & Jaeger, 1959; Metzger et al., 2004; Molina-Giraldo et al., 2011).
T Q q x D c c r q D c t w w r tDt t w w
4 2 2 16 4 2 2 exp exp ² ² ² ²²c d 2 (7b) This equation can also be expressed using the Hantush Well-function W (α, β) (Hantush, 1956):
Δ exp , 4 2 t w w Q q x c T W D c (7c) where 2 4 t r tD (8) 2 t w w r q c D c (9)
,
1exp 2 d 4 W (10) The MILS model (Equation 7a–7c) can be used to model and reproduce the temperature increase induced by a line heat source in an infinite medium. Considering this, Diao et al. (2004) described the behavior of the temperature increase under different flow conditions and demonstrated the impact of groundwater flow on performances of geothermal heat exchangers. They showed that the temperature increase is first mainly controlled by heat conduction for short times and then controlled by heat advection that dissipates the heat produced, which limits the temperature rise and leads to the stabilization of temperature for late time. For steady-state conditions (t → ∞), they proposed to approximate Equation 7b using the Bessel function of second kind and order zero K0:0 Δ exp 2 2 w w 2 w w f t t Q q x c rq c T K D c D c (11) where ΔTf is the temperature reached for steady conditions. Diao et al. (2004) demonstrated that the
dura-tion ts required to reach steady state, i.e., ΔT 0.99 ΔTf, directly depends on the magnitude of the ground-water flux q: 2 4 ² ² s w w c t c q (12) For short times, advection can be neglected, which means that the temperature increase can be modeled using the ILS model (Equation 5) considering only heat transfer by conduction. Since the behavior of tem-perature for steady-state conditions can be approximated by Equation 11, it is possible to define the intersec-tion time ti between the conduction-dominant stage, described by Equation 5, and the advection-dominant
stage, described by Equation 11:
0 1ln 2.25 exp 2 ²i t 2 t w w 2 t w w t D q x c K rq c r D c D c (13) By using the approximation e K xx 0
K x0
2.3log 0.8910
x
(Hantush, 1956), the time ti can beex-pressed as 2 2 2 2.24. i w w c t q c (14)
2.1.4. Applications for Active-DTS Experiments
The ILS and MILS models have been used to model temperature increase for different active-DTS applica-tions. Bakker et al. (2015) and des Tombe et al. (2019) combined a vertical line heat source with a FO cable and used the MILS model to successfully reproduce the temperature increase measured all along the FO cable and to estimate groundwater fluxes. However, they did not consider the possible variations of thermal conductivity of the materials along the FO cable, which may increase the uncertainty on the estimated fluxes.
The application of the MILS model when using a single FO cable as a heat source and for temperature measurement has been seldom proposed. Among others, F. Selker and J. S. Selker (2018) offered to use the definition of the stabilization time (Equation 12), initially developed by Diao et al. (2004), to estimate the groundwater flux. Del Val (2020) recently studied the effect of the FO cable on the temperature increase. She showed that the evolution of the temperature measured in the cable is the combination of the heat propaga-tion through the FO cable (ΔTFO) with the heat propagation through the porous media (ΔTPM):
ΔT ΔTFOΔTPM
(15) For very small times (t < 2 min), the temperature evolution is driven by heat propagation through the FO cable, described by the authors as “the skin-effect.” This effect is equivalent to the wellbore storage effect in groundwater hydraulics and depends on cable properties (diameter and thermal properties of the cable it-self). It induces a relatively large increase of temperature ΔTFO during the early period of heating due to the
thermal properties of the FO cable. Considering the effect of the cable, she proposed a graphical interpreta-tion of the flux based on the identificainterpreta-tion of the intersecinterpreta-tion time (Equainterpreta-tion 14). However, the application of the method to a field case remains uncertain considering the error on flow estimation induced by noise in the data and the difficulty in removing the skin-effect and in defining ti.
Besides these promising applications based on the use of the MILS model to interpret the temperature increase measured during active-DTS experiments, the method suffers from the lack of a standardized and validated interpretation method. In order to develop such a method, we present next a numerical model which considers all thermal processes controlling the temperature increase measured along a heated FO cable, namely heat transfer processes occurring through both the FO cable and the surrounding media. The aim is to highlight the effects of the FO cable, of the thermal properties of the media, and of the ground-water flow on the temperature rise in order to improve the understanding of the different thermal regimes controlling the thermal response over time. Numerical simulations will be used as a guide for developing and testing a theoretical interpretation methodology based on MILS models. This approach will be then validated using experimental data conducted during laboratory tests in a sandbox.
2.2. Numerical Model
As shown in Figure 1, the numerical model considers a simple 2D domain, within which heat transfer oc-curs with steady-state fluid flow in porous media. Simulations were done using the Conjugate Heat Transfer module of COMSOL Multiphysics®. Mesh size ranges from 4.85 × 10−6 to 2.68 × 10−2 m with the finest
meshes around the FO cable. In order to simulate the thermal response for typical groundwater fluxes and to define the limitations of the method, the thermal response was modeled under a large range of flux rang-ing from 8 × 10−7 to 3 × 10−1 m/s. Defining the limitations of the method requires ensuring the stabilization
of temperature expected under flow conditions, which involves long heating periods, especially for very low flow. After many tests, the heating period was set to 225 h, which necessarily involves the propagation of heat over a large domain whose dimensions depend on the groundwater fluxes. Thus, the size of the domain was adjusted for each flux tested to ensure that the heat produced does not reach the boundary of the do-main (Thermal Insulation Boundaries). Under no-flow conditions, the dodo-main was modeled as a square of 3 × 3 m with the heat source located in the center of the model. Under flow conditions, the domain is a rec-tangle whose size varies according to the flux: the increase of the flux requires increasing the length of the domain and reducing its width. For instance, the size of the model was set at 5 × 0.20 m for q = 3 × 10−5 m/s
and the heat source was located 4 cm of the laminar inflow boundary condition. For q = 3 × 10−6 m/s, its
For each simulation, the size of the domain was validated by verifying the stability of the temperature near the boundaries for the whole heating period. A single and large domain, for instance, 5 × 5 m could have been used but it would have resulted in more significant run-time simulations, especially with the extra fine mesh size of the model. Considering the mesh size of the model as a characteristic length (COMSOL Mul-tiphysics, 2019), the Courant number ranges between 2.88 × 10−4 and 0.62 and the Peclet number ranges
from 5 × 10−3 to 0.91; such values lower than 1 ensure numerical stability.
The FO cable was explicitly implemented with a steel core and a plastic jacket to get a better understand-ing of the effect of the cable design (material, layer thickness, conductivity of materials) on the thermal response, following Read (2016). The plastic jacket and the steel core radius are defined in accordance with the cables used for field experiments, namely BRUSens cables (reference LLK-BSTE 85°C) whose external and steel tube diameters equal, respectively, to 3.8 and 2.2 mm. This simplified representation of the cable does not include all of its components, such as fiber optics or gel. Still, due to the high thermal conductivity of the steel, this representation is sufficient to reproduce heat transfer processes (Read, 2016), and thus the thermal response induced by the heated cable in the saturated porous medium. The active-DTS heat tracer experiment was simulated by applying a heat source term (W/m3) in the steel core, defined as the ratio
be-tween the electrical power input (W/m) and the surface of the steel core (m2).
Temperature variations throughout the domain and in the cable core were first modeled under no-flow condition, for q = 0 m/s, with heat transfer being exclusively controlled by conduction. Then, simulations were conducted by sweeping the groundwater flux over a range of 10−17 to 3 × 10−1 m/s. All simulations
were performed using two values of thermal conductivity for the porous media, i.e., λ = 1.1 and 3.5 W/m/K, which allows covering a large range of thermal conductivities commonly found for sedimentary aquifers (Stauffer et al., 2013). Durations of heating periods were adjusted according to flow to ensure temperature stabilization at late times. Temperature distributions during recovery, after the end of heat injection, were also simulated to obtain the evolution of temperature during both heating and recovery periods.
2.3. Sandbox Laboratory Experiments 2.3.1. Experimental Setup
Laboratory experiments were conducted in a sandbox in which flow rates can be well controlled in order to test the proposed active-DTS interpretation method on experimental data. As shown in Figure 2, active-DTS experiments have been conducted by deploying a FO cable in a 0.576 m3 PolyVinyl Chloride tank open at
the top (1.6 m long, 1.2 m width, and 0.3 m height) and filled with 0.4–1.3-mm-diameter quartz sand that comprises 12% of fine sand, 28% of medium sand, and 60% of coarse sand. The height of water in reservoirs on two sides of the sandbox (h1 and h2) can be adjusted manually to control of hydraulic gradient and thus
the water flow through the sand. The setup of the sandbox is such that flow can be considered homogene-ous. The average hydraulic conductivity was estimated at 3.06 × 10−3 m/s (Table 1). This experiment was
Figure 1. Numerical model geometry: the model simulates heat transfer by conduction with steady-state fluid flow
using the Conjugate Heat Transfer module of COMSOL Multiphysics®. The thermal source (located at x0 and y0) was represented by a simplified FO cable considering the effect of the cable design on thermal storage and conduction. FO, fiber optic.
the subject of a precedent study whose aim was to discuss the feasibility of conducting active-DTS meas-urements in a sandbox knowing that the spatial resolution of DTS units could be close to the size of the experiment. More details about the experimental setup are provided in Simon et al. (2020).
The cable was buried inside the sandbox at 20 cm-depth below the sand surface. The cable was installed so that the angle between the flow and the FO cable varies in space. Cable sections called DE, HG, and HI are perpendicular to water flow (90°), whereas cable section CD is parallel to water flow. Sections EF and FG lie, respectively, at a 60° angle (α) and a 110° angle (β) to the water flow direction. The FO cable used is a 3.8-mm-diameter cable containing four multimode 50/125-µm fibers (BruSens cable; reference LLK-BSTE 85°C). As shown in Figure 2, two fibers were spliced at the end of the cable, allowing measurements in double-ended configuration (van de Giesen et al., 2012). The paired fiber was connected to a Silixa Ultima S DTS unit reporting temperature every 12.5 cm at a-20 s sampling interval (10 s per channel). The effective spatial resolution of the unit was experimentally estimated during heating periods to be between 51 and 67 cm (Simon et al., 2020). To calibrate the DTS unit, 20 m of cable was placed in a warm calibration bath and 20 m in a cold calibration bath (a box filled with wetted ice). In each bath, a PT100 probe (0.1°C) and a RBR SoloT probe (0.002°C accuracy) recorded the temperature to calibrate the acquired data. With this setup, the relative uncertainty of measurement was estimated to 0.03°C while absolute uncertainty of meas-urement can be estimated at 0.15°C. External temperature probes, marked from 1 to 8 in Figure 2, allowed monitoring temperature variations in the sandbox at different distances from the FO cable.
2.3.2. Heat Tracer Experiments
A 7-m section of cable was electrically isolated and connected to an electrical cable (connections C1 and C2)
to allow the injection of electricity from a power controller. A succession of active-DTS experiments was conducted under different flow rates for 4 days to characterize the effect of flow on the thermal response. As shown in Table 1, the recovery period following each heating was monitored at least 14 h after turning off the power controller, ensuring the return to the ambient temperature. The flow rate through the sandbox
Figure 2. Experimental setup showing the dimensions of the sandbox and the deployment of FO cable (top view [x,y
plane] and side view [x,y,z plane]). The marks C1 and C2 represent the electric connections between the FO cable and the electrical cables allowing the injection of electricity into the steel frame of the FO cable (modified from Simon et al., 2020). FO, fiber optic.
was progressively decreased, until the no-flow condition was reached (q = 0). For each experiment, the flow rate was measured independently and manually at the outlet of the bench. The hydraulic conductivity was calculated using Darcy's law under unconfined conditions. An estimate of the water flux according to the distance from the inlet reservoir (h1) was also calculated (Domenico & Schwartz, 1998). Table 1 provides
details about flow conditions for each day of the experiment and the estimate of the hydraulic conductivity and water flux for section DE (located 20 cm from the inlet reservoir). For each experiment, the cable was energized continuously during a few hours (between 6 and 8 h) using a Silixa Heat Pulse Control System, delivering a well-controlled power intensity of 20 W/m along the heated section, by injecting continuously 141 W in the electrical circuit with an input amperage of 7.4 A.
Considering the spatial resolution of the DTS device and the limited size of the sandbox, the feasibility of conducting active-DTS experiments in such conditions was discussed in a previous publication (Simon et al., 2020). We validated experimental results for most measurement points, except section GH that was highly affected by heated adjacent sections of cable. In addition, we have ensured and verified that the boundaries of the tank were not affected by the heat injection along the FO cable.
3. Results
3.1. Numerical Simulations Results
Figure 3a shows the evolution of the simulated temperature during heating under different flow conditions. The log-time derivative of temperature is shown along the temperature evolution to highlight the effect of the different processes following Renard et al. (2008). During the first seconds of heat injection through the steel armoring of the cable, the evolution of both the temperature and its derivative is similar independently of the water flux. The temperature increases very rapidly and ΔT reaches 16.8°C in about 100 s. After few seconds, the derivative departs from the temperature curve and makes a hump. Afterward, the thermal re-sponse depends on flow conditions. When heat transfer is exclusively by conduction (q = 0 m/s), a gradual and continuous increase of temperature is observed over time and the log-time derivative tends toward a constant value of 2.51 (solid and dotted red lines).
Under flow conditions (green, orange, and blue lines), the temperature stabilizes progressively at late time. The associated log-time derivative decreases continuously before stabilizing toward zero. The value of ΔT after temperature stabilization depends on groundwater flux. As expected, the higher the water flux, the lower the value of ΔT. For each example, characteristic times td and ta are defined and highlighted, since
these times allow a proper characterization of q as we will see in the following sections. Time td corresponds
to the moment when the temperature curves under flow conditions separate from the curve under no-flow conditions and when the associated log-time derivatives start decreasing. Time ta corresponds to the
mo-ment when temperature stabilizes and the log-time derivative reaches zero. As shown in Figure 3a, these two characteristic times depend also on the magnitude of flux.
The analysis of the behavior of both the temperature and its log-derivative in the semilogarithmic plot appears efficient to delimitate the effect of the different parameters on the temperature increase. Thus, by analogy with well-test interpretation, the hump made by the log-derivative in early times can be interpreted
Day 1 Day 2 Day 3 Day 4
Experimental conditions Flow rate (m3/s) 2.12 × 10−5 1.33 × 10−5 4.8 × 10−6 0
Duration of heating period 04h20m 08h00m 08h05m 08h00m
Injected power (W/m) 20
Duration of recovery period (h) 15 15 14 20
Calculated parameters Hydraulic conductivity (m/s) 3.12 × 10−3 ± 1.29 × 10−4 3.24 × 10−3 ± 2.54 × 10−4 3.06 × 10−3 ± 7.98 × 10−4 Flux (Darcy velocity) (m/s) along
section DE 4.93 × 10
−5 2.95 × 10−5 1.11 × 10−5 0
Table 1
as the result of “storage effects” (Del Val, 2020; Papadopulos & Cooper, 1967; Renard et al., 2008), which clearly helps to delimitate the period when temperature increase is affected by the FO cable. Moreover, temperature stabilization, which should be reached once the heat injected through time is fully dissipated by heat advection, could be investigated through the log-derivative similarly to case of leaky aquifers or constant head boundary aquifers that are identified through the behavior of drawdown evolution (Han-tush, 1956; Hantush & Jacob, 1955; Renard et al., 2008).
Figure 3b shows the evolution of both temperature and log-time derivative of simulated temperature during heating periods under different flow conditions and for different porous medium thermal conductivities. During the first 20 s of heat injection, the responses are fully independent of both flux and thermal con-ductivity. Afterward, temperature increases independently of flow for a few minutes. During this period, the temperature increase is higher for a smaller thermal conductivity. At later times, the thermal response depends on both thermal conductivity and flow that dissipates the heat produced, leading to temperature stabilization. The higher the thermal conductivity, the lower the value of ΔT.
3.2. Description of the Typical Thermal Response Curve
Numerical simulations highlight the role of the distinct heat transfer processes on the temperature rise dur-ing active-DTS experiments. These numerical results will provide the basis for the proposed general inter-pretation of thermal response curves and allow the definition of the typical response curve expected under flow conditions during active-DTS experiments using the FO cable as the heat source. Figure 4a illustrates this typical response and indicates the heat transfer processes at play in different stages of the temperature rise during heating.
As shown in Figure 4a, the temperature increase during the experiment is the result of the combination of the effect of the FO cable (ΔTFO) and the heat transfer occurring through the porous media (ΔTPM). ΔTFO
is the result of the heat storage and heat conduction occurring through the FO cable from the beginning of the heat injection. The hump made by the curve of the log-time derivative shows that these processes affect the temperature increase only during a short period, from the start of heating to time tc. As soon as the heat
produced reaches the surrounding material, thermal heat processes occurring through the porous media begin to control the temperature increase. After a very short time, the temperature increase depends on the
Figure 3. Typical examples of thermal responses obtained from active-heating experiments simulated using the
numerical model described in Section 2.2. These curves highlight (a) the effect of groundwater flux on both the temperature (solid curves) evolution and its derivative (dotted curves) and (b) the effect of thermal conductivity and flow conditions on both the temperature evolution (solid curves) and its derivative (dotted curves).
thermal properties of the porous media (as demonstrated in Figure 3b). For a short period, the evolution of the temperature is thus simultaneously controlled by heat conduction and storage in the cable as well as heat transfer occurring in the porous medium (conduction-affected stage).
For t > tc, temperature variations are no longer sensitive to the effect of the heated FO cable and the
temper-ature rise is exclusively controlled by heat conduction through porous medium in accordance with Equa-tion 5. This gradual increase of temperature observed from t = tc up to t = td corresponds to the
“conduc-tion-dominant period.” When t exceeds td, the temperature rise departs from the conduction regime with a
decreasing rate of temperature rise. Time td can be defined as the limit between the conduction-dominant
stage and the advection-dominant stage, which should occur for a thermal Peclet number Pe = 1. A transi-tion period is then observed between td and ta, after which time the temperature stabilizes at the maximum
temperature ΔT(t) = ΔTf. After t = ta, advection is clearly dominant and the heat injected is fully dissipated
by advection and conduction in the porous medium.
Further insights on the thermal response is provided in Figure 4b, which shows the temperature variation reached after a specific heating time t, ΔT(t), depending on groundwater flux. Figure 4b clearly shows that for a given thermal conductivity, a broad range of fluxes, between qmin and qmax, can be estimated from the
temperature rise ΔTf. For q < qmin, advection is negligible and conduction dominates heat transfer, the
tem-perature response being insensitive to flow. For q > qmax, the temperature rise is limited to the temperature
increase induced by the FO cable, ΔTFO, most of the heat produced being dissipated by advection except the
one stored within the FO cable. It should be noted that the typical response curve (orange curve in Figure 4a) can be observed between qmin and qc (orange hatched area). Between qc and qmax, the conduction-dominant
period is negligible, meaning that ΔTPM is entirely controlled by advective heat transfers occurring through
the porous media. Despite that, between qmin and qmax, each value of q can thus be associated with a single
value of ΔTf. The uncertainty associated with groundwater flux estimates can be also easily deduced from
Figure 4b, knowing the accuracy of temperature measurements.
Figure 4. (a) Theoretical thermal response curves showing the effect of heat transfer processes on the temperature increase measured during heating. The
orange curve describes the typical response curve expected during an experiment. The black and brown curves correspond to the behavior expected for extreme and specific hydraulic conditions, respectively when q → 0 and for high groundwater flux. (b) Schematic evolution of the value of ΔTf(t) depending on groundwater flux after a given heating time t.
3.3. Interpretation of the Thermal Response Curve 3.3.1. The Use of the MILS Model
The behavior of the temperature evolution through time described in the previous section has important consequences for the interpretation of temperature data obtained from active-DTS experiments. The ana-lytical solution from Des Tombes et al. (2019) cannot be used directly here since the increase in temperature associated with heat conduction within the FO cable is not considered. As shown in Figure 3, this difficulty is due to the fact that the temperature increase during the first stage of the thermal response is simulta-neously controlled by heat transfers occurring through the FO cable and by heat conduction occurring through the porous medium. In the following, we propose an interpretation method that accounts for the effect of the FO cable and provides independent estimates of the varying thermal properties of the porous media and the groundwater flux.
As shown in Figure 5, the interpretation focuses on the second part of the thermal response curve, when the temperature increase is only controlled by heat transfer occurring through the surrounding porous me-dia and thus driven by heat conduction and advection, using the MILS model. First, the approach requires the identification of time tc corresponding to the duration of the first stage of the thermal response, during
which heat conduction occurs mainly through the FO cable. This really short delay (around 50 s here) is marked by the end of the hump made by the log-time derivative of temperature and can be graphically de-fined both by the shape of the log-derivative of temperature and by the fact that the conduction-dominant period should be apparent as the start of the straight line increase in temperature from t = tc (straight line
s1 in Figure 5).
Once tc is defined, the MILS model (Equation 7a–7c) can first be used by considering q = 0 m/s (gray
line) to reproduce the temperature increase occurring during the conduction-dominant period (i.e., for
tc < t < ∼1,000 s in Figure 5). Here, the temperature increase during the conduction-dominant period
can be perfectly reproduced by the analytical solution Root-Mean-Square Error (RMSE) < 0.01°C) by ap-plying a value of λ equal to 1.1 W/m/K, in perfect agreement with the thermal conductivity considered in the numerical model. Thus, this approach can provide a direct estimation of the thermal conductivity of the porous media. Knowing the value of thermal conductivity, the temperature increase can be then modeled for any t > tc using the MILS model considering both the previous value of thermal
conductiv-Figure 5. Application of the MILS model to reproduce the temperature increase observed during the second stage
of the thermal response curve. In this example, the approach was tested using one of the thermal response curves numerically simulated (black line) considering λ = 1.1 W/m/K and q = 1 × 10−5 m/s (see Section 2.2). The analytical solution can be used to reproduce the temperature evolution (i) during the conduction-dominant period only by considering q = 0 m/s and λ = 1.1 W/m/K (gray line) and then (ii) during the advection-dominant period (red dotted line) by considering the effective value of the flow (q = 1 × 10−5 m/s) once the thermal conductivity is defined. Then, the extrapolation of the MILS model for t < tc provides an estimate of ΔTFO, and therefore of ΔTPM. MILS, Moving Instantaneous Line Source.
ity and the unknown value of groundwater flux. Here, by considering
q = 1 × 10−5 m/s (dotted red line in Figure 5), the MILS model reproduces
perfectly the temperature evolution during both the conduction-domi-nant and the advection-domiconduction-domi-nant periods (RMSE ≈ 0.02°C). Thus, this simple approach can provide a direct and precise estimation of both the thermal conductivity of the porous media and the flux. By deduction, this approach also provides a direct estimation of the effect of the cable ΔTFO
on the thermal response whose direct quantification remains otherwise difficult.
3.3.2. Graphical Interpretation of the Thermal Response Curve
In this section, we propose another complementary approach to inter-pret the temperature evolution through time by defining characteristic times related to groundwater flux. As shown in Figure 6, under flow con-ditions (orange line), the different heat transfer regimes can be easily de-fined and delimited by well-marked changes in temperature evolution. The change in temperature is linear in the semilogarithmic plot under both the conduction-dominant period and the stabilization-time period, which are respectively characterized by straight lines with slopes s1 and
s3. The temperature trend during the transition-time period can also be approximated by a slope s2. Again, it
is important to clearly define the transition period to avoid any misinterpretation with the conduction-dom-inant period and to ensure a correct determination of the characteristic times.
As shown in Figure 6, the transient-flow method for determining the thermal conductivity developed by Blackwell (1954) (Equation 6) can be applied to determine slope s1 under the conduction-dominant period.
The duration of the conduction-dominant period is variable and depends on both thermal conductivity and groundwater flow. The slope should be estimated from the end of the hump of the log-derivative to the start of temperature stabilization, marked by the slope change to s2. The numerical simulations
pre-sented in Figure 3 were used to graphically estimate the thermal conductivity using the calculated slope
s1. The thermal conductivity was exactly estimated for each simulation, with a mean standard deviation of
0.07 W/m/K. Under no-flow conditions, the derivative tends toward the value of the slope s1 (2.51 on the
example in Figure 6). Under flow conditions, the estimation is less accurate for high fluxes (±0.2 W/m/K when q = 3 × 10−5 m/s and ±0.03 W/m/K when q = 3 × 10−6 m/s) because the conduction-dominant period
is shorter and therefore more difficult to delimitate. Note also that the use of this approach requires that Equation 3 can be approximated by the Well-function, i.e., that r2/4D
tt << 1 (Blackwell, 1954; de
Marsi-ly, 1976). Here, considering the very small diameter of the FO cable (r), this condition is satisfied as soon as
t is greater than a few seconds (for λ = 2 W/m/K and ρc = 3 × 106 J/m3/K).
In addition, it is possible to use the intersection time ti defined as the intersection between the
conduc-tion-dominant period and the advecconduc-tion-dominant period to estimate groundwater flux (Equation 14). This time can be easily estimated graphically when both slopes s1 and s3 are well defined and when the transition
period is clearly identified. Similarly, the flux may be estimated from characteristic time td, which
corre-sponds to the intersection point between slopes s1 and s2. For t = td, the thermal Peclet number should equal
1 so that 1 t dq Pe D (16) with d the characteristic length that can be represented by the mean grain size (Stauffer et al., 2013). By assuming that the characteristic length at t = td corresponds to the diffusion length, d can be defined as
2 t d d D t leading to 2 dt D q t (17)
Figure 6. Graphical interpretation of the thermal response: definition
of the characteristic times td, ti, and ts to estimate groundwater flux and calculation of slope s1 during the conduction-dominant period to evaluate thermal conductivity.
Lastly, the groundwater flux can also be defined through the steady-state time ts reached when ΔT 0.99 ΔTf (Equation 12) (Diao et al., 2004; F. Selker & J. S. Selker, 2018).
To summarize, the groundwater flux can be estimated graphically by defining the three characteristic times
td, ti, and ts: 2.24. 4 2 d i w w² ² s w w² ² c c q c t t c t c (18) The equivalence below may be verified to validate the estimation of characteristic times:
1.786
s i
t t
(19) One of the main advantages of this method is that these times are completely independent of the effect of the FO cable on temperature rise. Values of groundwater flux were estimated for each numerical simula-tion using the graphical approach by determining the characteristic times td, ti, and ts. The application of
the graphical method provides a very good estimate of the flux with a mean error of around 4.7%. The error on the estimation increases with the increase of flux magnitude and reaches 7.4% when q = 3 × 10−5 m/s
against 1.3% when q = 3 × 10−6 m/s.
3.3.3. Expected Increase of Temperature
The analytical solution can also be used to define theoretical curves ΔTPM = f (q, λ, t) that can relate each
value of ΔTPM with a value of flux q, estimated for a specific heating duration t, and a given thermal
conduc-tivity λ, as illustrated in Figure 7. For a given heating duration once λ is estimated, such curves can provide a direct estimation of the flux if the temperature increase due to the FO cable (ΔTFO) has been previously
estimated. These curves can also help to define more precisely the limits of the method and especially the values of qmin and qmax, beyond which the flux estimate is not possible. Note that, in accordance with
Equa-tions 7, the temperature increase is proportional to the injected power Q. Thus, these theoretical curves can be established for either a specific value of power, ΔT = f (q, λ, t, Q), or normalized, ΔT/Q = f (q, λ, t). In practice, the normalization of the temperature increase by the injected power can help to compare results conducted under different conditions and verify the reproducibility of measurements.
3.4. Experimental Validation 3.4.1. Experiments Results
Figure 8a shows the evolution of the mean temperature measured along section DE during the experiment, which constitutes the ideal case in which the cable is perpendicular to flowing water (Figure 2). Considering the flow direction and the transport of heat generated along the cable from upstream to downstream (from
Figure 7. Theoretical curves ΔTPM = f (q, t = 24 h) showing the expected value of ΔTPM depending on groundwater flux and thermal conductivity.
DE to IJ), section DE is the only one whose temperature evolution is not affected by upstream heated sec-tions. The representativeness of DTS measurements along this section was validated by Simon et al. (2020) who studied the effective spatial resolution during active-DTS experiment using the same data set as the one used in this study. The value ΔT corresponds to the difference between the temperature measured over time and the initial temperature. For each experiment, the start of the heating period (yellow periods in the top of Figure 8a) and the start of the recovery period (blue periods in the top of Figure 8a) induce rapid and sharp temperature changes (at least 12.5°C in less than 3 min). As expected, the lower the flow rate or flux, the higher the ΔT reached at the end of the experiment. Note also that the inflow temperature changed slightly over time, mainly due to the loop water circulation, affecting the thermal response. Thus, data were postprocessed by applying a filter based on the propagation and the attenuation of temperature changes inside the sandbox using classical heat transport equations (Anderson, 2005; Domenico & Schwartz, 1998) in order to keep only the temperature variations induced by the heat injection (Simon et al., 2020).
Figure 8. (a) Succession of active-DTS experiments conducted under different flow conditions and evolution of
the mean temperature increase, ΔT, recorded along section DE during heating and recovery periods. (b) Evolution of the temperature increase (solid lines) and the log-time derivative of temperature (dotted lines) for a flow rate of 4.8 × 10−6 m3/s (red lines; Day 3) and under no-flow conditions (black lines; Day 4).
Figure 8b shows the evolution of temperature and its log-time derivative measured under no-flow (black lines) and flow conditions (4.8 × 10−6 m3/s, red lines). During the first 30 min of heat injection, the
temper-ature evolution is similar for both curves independently of flow conditions. The tempertemper-ature increases very rapidly in the first 2 min with ΔT reaching 12°C. Then, the temperature keeps increasing gradually with ΔT reaching 13.9°C after 30 min of heat injection. Afterward, the temperature evolution under flow conditions (red lines) departs from the temperature evolution under no-flow conditions and stabilizes progressively around 14.5°C, meaning that steady state is reached. Concerning the evolution of the log-derivative of temperature (dotted lines), the derivative makes a hump at very short times and seems to stabilize for times greater than 2 or 3 min.
Thus, the temperature evolution with time measured along the section DE seems in very good agreement with the expected behavior. Results show (i) a gradual and continuous increase of temperature under no-flow conditions (Carslaw & Jaeger, 1959); (ii) the effect of advection on the late thermal response under flow conditions (Carslaw & Jaeger, 1959; Diao et al., 2004; Zubair & Chaudhry, 1996); (iii) that higher values of ΔT can be associated with lower flow velocities (Bakx et al., 2019; Perzlmaier et al., 2004), and (iv) the effect of heat storage within the cable is limited to the early moments of heating, as highlighted by the char-acteristic behavior of the derivative (Del Val, 2020). To go further, these promising data will be used in the following sections to apply and validate the two interpretation methods previously described, i.e., the MILS model and the graphical approach.
3.4.2. Detailed Application of the Interpretation Methods
Figure 9 provides an example of the interpretation of the thermal response curve obtained during the sec-ond day of the laboratory experiment (q = 2.95 × 10−5 m/s). Figure 9a focuses on the use of the MILS to
reproduce the temperature evolution during the second stage of the curve (t > tc). The first step of the
interpretation workflow consisted in defining the time tc, from which the temperature increase is
exclu-sively controlled by heat transfer through the porous medium. In this example, tc is around 110 s, which
corresponds to the end of the hump made by the log-time derivative of temperature and the start of the conduction-dominant period.
Figure 9. (a) Interpretation of the thermal response curve (red solid line) obtained during the second day of
the experiment (q = 2.95 × 10−5 m/s). The dotted red line represents the log-time derivative of temperature. The temperature rise was matched with the MILS model (black lines) (Equation 7a) for q = 0 m/s (for tc < t < 600 s) and
q = 2.95 × 10−5 m/s (for t > t
c). In the MILS model (Equation 7a), the distance (x,y) between the heat source and the measurement point was fixed to x = 1.9 × 10−3 m and y = 0 m, considering flow in the y direction, with x the effective radius of the FO cable used during the experiment and measured using an electronic caliper. (b) Application of the proposed graphical method based on the recognition of three characteristic times allowing the estimation of groundwater flux. The percentage of error of q estimates obtained from the three characteristic times compared to the experimental measurement of q is indicated. The graph highlights the last part of the temperature curve to better show characteristic times. MILS, Moving Instantaneous Line Source; FO, fiber optic.
In the next step of the interpretation workflow, the MILS model considering q = 0 m/s (Equation 7a rep-resenting a line in the semilog graph) was used to estimate the effective value of thermal conductivity for
tc < t < 600 s. For that purpose, the MILS model was used with a range of values of thermal conductivity
(from 1 to 4.5 W/m/K with a step of 0.01 W/m/K). For each value of thermal conductivity, the RMSE be-tween the predicted model and the measured temperature during the conduction-dominant period was calculated. The optimal value of the estimated thermal conductivity with the lowest RMSE (0.04°C) was found at λ = 2.96 W/m/K (black line in Figure 9a).
For the following step of the interpretation workflow, the flux q can be estimated by varying q in the MILS model (Equation 7a) with the previously estimated value of thermal conductivity. The optimal value of q is the one leading to the lowest RMSE between the MILS and the observed temperatures from the start of the conduction-dominant period (t > tc) until the end of heating. For the experimental results, the previously
estimated thermal conductivity (λ = 2.96 W/m/K) was used in the MILS model together with the value of
q (2.95 × 10−5 m/s) estimated experimentally. With this value of q, the MILS model perfectly reproduced
the experimental data with a RMSE between the experimental data and the MILS model around 0.1°C (Figure 9a).
The application of the MILS model with the optimal values of thermal conductivity λ and flux q also allows the estimation of the two parts of the active-DTS temperature rise respectively related to the porous media ΔTPM and to the FO cable ΔTFO (Figure 9a). Although it occurs over a very short period, the effect of the
FO cable properties can thus be experimentally observed through the behavior of the log-time derivative of temperature (dotted red line in Figure 9a). The data interpretation can be also based on the evolution of the log-time derivative of temperature for later times (t > tc) which is however more difficult to achieve since the
calculated derivative is noisy. The noise in the derivative is greatly dependent on the temperature resolution and sampling time of the DTS device. This point is a classical issue when calculating the derivative of exper-imental data, especially for well-test interpretation (Ramos et al., 2017; Renard et al., 2008).
As shown in Figure 9b, the graphical approach was similarly applied to estimate both the thermal con-ductivity and the flux. The temperature evolution at intermediate and late times was approximated by the three different slopes s1, s2, and s3. The value of the slope of each interval that can be defined within the
conduction-dominant period by varying the duration (5 points minimum) and the bounds of the considered interval was calculated. The average and standard deviation of slope estimates with R2 > 0.98 was
calcu-lated, providing a rough estimate of the thermal conductivity of the saturated porous media (Equation 6), equal to 2.89 ± 0.32 W/m/K, in very close agreement with the previous estimate. Then, characteristic times
td, ti, and ts (Figure 9b) were defined to provide independent estimates of flux (Equation 18), which are also
in very good agreement with the expected value from the laboratory measurement. Note that values of the volumetric heat capacity of water (ρwcw) and of the porous medium (ρc) were fixed at 4.1 × 106 and 3 × 106 J/
m3/K, respectively. The effect of the volumetric heat capacity of the porous medium will be discussed next.
3.4.3. Interpretation of the Different Experimental Thermal Curves
As shown in Figure 10, the two proposed interpretation methods were also applied to the temperature evo-lution measured for each day of the experiment described in Section 3.4.1.
Figure 10a shows that the thermal conductivity was graphically estimated for each experiment (gray lines) by calculating the slope during the conduction-dominant period. Under no-flow conditions (dark blue line), the thermal conductivity is evaluated at 2.89 ± 0.2 W/m/K. This result is consistent with the common range of thermal conductivity of saturated sands, varying from 1.5 and 4 W/m/K (Stauffer et al., 2013) and can be considered here as the actual thermal conductivity of the saturated sand. Under flow conditions (green, red, and light blue lines), the duration of the conduction-dominant period is shorter for higher fluxes. The esti-mated values of thermal conductivity are still very close from the reference value. Of course, the uncertainty is higher for higher fluxes (σ = 0.45 W/m/K for the highest flux) because of the shorter duration of the conduction-dominant period. The graphical estimation of flux based on the determination of characteristic times also provides very good results for the different experiments, with a maximum error of 6%. Results for the first and the third experiments are not shown since they are very similar to the examples described in the previous section.
Figure 10b shows the application of the MILS model (Equation 7a) to reproduce the thermal response under the different flow conditions. For each case, the model was applied for t > tc, when the temperature
increase is fully independent of the effect of the FO cable. Modeled curves fit very well with the experi-mental curves. The RMSE calculated between observed data and the analytical model varies between 0.03 and 0.06°C. To validate the MILS model, the rise in temperature monitored inside the sandbox 5 cm from section DE was also modeled using the MILS model. At a 5-cm distance from the FO cable, the temperature increased by 1.75°C after a 5-min-delay, corresponding to the time required for the advective transport of heat from the heated FO cable to the probe. As shown in Figure 10b, this temperature increase can be very well reproduced using the analytical solution knowing the distance between the FO cable and the probe and considering λ = 2.96 W/m/K and q = 1.11 × 10−5 m/s. This result validates the use of the analytical solution
and strengthens the estimate of flux.
Lastly, as shown in Figure 10c, the modeled ΔT after a given heating period (t = 4 h) can be calculated as a function of flux by summing the effect of the cable ΔTFO, graphically determined, with the temperature
increase induced by heat transfer through the porous media ΔTPM, estimated using the MILS model.
Ex-perimental data points have been reported on this curve and perfectly fit on the curve obtained from the MILS model, thus providing another validation of the applicability of the proposed interpretation method based on the MILS model. The value of ΔT (at t = 4 h) can thus directly be used to estimate fluxes ranging
Figure 10. (a) Estimation of thermal conductivity under the different flow conditions using the slope s1 during the conduction-dominant period. For each flux, the slope used to estimate thermal conductivity λ is shown. (b) Use of the MILS model to reproduce thermal responses measured along section DE under the different flow conditions. The analytical solution can be used (i) to reproduce each thermal response curve and (ii) to reproduce the temperature evolution measured with an external probe (PT100 no. 2; see Figure 2) located 5 cm downstream of section DE during the third heating period (q = 1.11 × 10−5 m/s). (c) By fixing λ = 2.9 W/m/K and ΔT
FO = 9.5°C, the theoretical curve ΔT = f(q), at t = 4 h of heating, can be defined, allowing the direct association of a value of ΔT with a value of flux. MILS, Moving Instantaneous Line Source.
between 8 × 10−6 and 3 × 10−2 m/s. An experimental error of 0.2°C (when estimating the value ΔT FO or
associated with noise measurement) would induce an error of only 10% on the estimated flux. Compared to typical uncertainties associated with the estimation of hydraulic conductivity, the proposed interpretation method of active-DTS experiments provides a much better estimate of groundwater flux (Darcy velocity), with a very low uncertainty.
4. Discussion
Many active-DTS experiments were previously conducted in different reservoirs (soil, sediments, aquifer, etc.) to estimate groundwater flux and/or thermal conductivity. Here, we provide a theoretical and general framework to interpret the thermal response during active-DTS experiments. Our study contributes to a better understanding of the different thermal regimes controlling the thermal response and in particular the respective role of thermal conduction and groundwater advection. Moreover, it shows very well that the temperature increase during the early times of the experiment is simultaneously controlled by heat transfer processes occurring through both the FO cable and thermal conduction in the porous media. To go further, we clearly highlighted the role of the different parameters in the different thermal regimes. Transitional behaviors were also precisely described improving the interpretation of experimental data.
As expected, the method proposed by Des Tombes et al. (2019), also based on the MILS model, is not direct-ly applicable when using a single heated FO cable. Indeed, the eardirect-ly times of ΔTPM (t < tc) cannot entirely
be reproduced using the MILS model since heat transfer processes through the FO cable also control the temperature increase during this period. However, once ΔTFO is estimated, the analytical solution can be
applied starting from t = 0 (start of the heat injection), since heat transfer through the FO cable does not delay the thermal response (the analytical solution considering the distance from the heat source). This implies that, for t > tc, the variations of temperature are similar regardless the configuration of the setup
(single cable or separated heat source/FO cable) are equal to ΔTPM and can consequently be reproduced
using the MILS model.
The improved and full understanding of the different thermal regimes controlling the temperature increase over time led us to propose two independent and complementary methods to interpret the thermal response, each providing an independent estimate of both thermal conductivity and groundwater flux. In addition, the first method that relies on the use of the MILS model considers the temperature increase induced by the FO cable allowing its quantification. The second method that depends on three complementary characteris-tic times is simple to apply and has the advantage to well define the limits of the different thermal regimes. Although both methods are complementary, the use of the MILS model is certainly more convenient to estimate the distribution of groundwater flows along a FO cable. The graphical approach requires a thor-ough analysis of the temperature evolution. The automated estimation of the characteristic times does not seem conceivable considering data noise, meaning that a “point-by-point” analysis is required. However, the graphical approach may be used to confirm some of the results obtained with the MILS model.
Both interpretation methods were successfully tested through numerical simulations and fully validated thanks to experimental data. The approach provides estimates of the thermal conductivity and the ground-water flux with an excellent and unprecedented accuracy. Experimental results show indeed that thermal conductivity can be easily estimated with a good accuracy around 0.1–0.2 W/m/K, which corresponds to ±2.5%–10% for typical values of thermal conductivity. This accuracy depends on the quality of data and groundwater flux but is as good as or better than estimates from Heat Pulse Probes or TRTs (He, Dyck, Hor-ton, Li, et al., 2018; He, Dyck, Horton, Ren, et al., 2018; Raymond et al. 2011). The accuracy of groundwater flux estimates is even better since our experimental results suggest it can reach a few percent which is much better than the one obtained with hydraulic methods (de Marsily, 1976). Moreover, while hydraulic meth-ods provide in general an integrated value of hydraulic conductivity, active-DTS experiments can provide continuous groundwater flow measurements with an excellent spatial resolution close to 1 m or less (Simon et al., 2020).
In the following sections, we discuss the sensitivity of the method to the different parameters and the ap-plicability of active-DTS experiments in the field. This provides some guidelines for users to achieve ac-tive-DTS experiments in different contexts.
